A unified mixed-integer programming model for simultaneous fluence weight and aperture optimization in VMAT, tomotherapy, and cyberknife

All rights reserved. In this paper, we propose and study a unified mixed-integer programming model that simultaneously optimizes fluence weights and multi-leaf collimator (MLC) apertures in the treatment planning optimization of VMAT, Tomotherapy, and CyberKnife. The contribution of our model is threefold: (i) Our model optimizes the fluence and MLC apertures simultaneously for a given set of control points. (ii) Our model can incorporate all volume limits or dose upper bounds for organs at risk (OAR) and dose lower bound limits for planning target volumes (PTV) as hard constraints, but it can also relax either of these constraint sets in a Lagrangian fashion and keep the other set as hard constraints. (iii) For faster solutions, we propose several heuristic methods based on the MIP model, as well as a meta-heuristic approach. The meta-heuristic is very efficient in practice, being able to generate dose- and machinery-feasible solutions for problem instances of clinical scale, e.g., obtaining feasible treatment plans to cases with 180 control points, 6750 sample voxels and 18,000 beamlets in 470 seconds, or cases with 72 control points, 8000 sample voxels and 28,800 beamlets in 352 seconds. With discretization and down-sampling of voxels, our method is capable of tackling a treatment field of 8000-64,000cm3, depending on the ratio of critical structure versus unspecified tissues.

Componentwise ultimate bounds for positive discrete time-delay systems perturbed by interval disturbances

This paper presents a method to derive componentwise ultimate upper bounds and componentwise ultimate lower bounds for linear positive systems with time-varying delays and bounded disturbances. The disturbance vector is assumed to vary within a known interval whose lower bound may be different from zero. We first derive a sufficient condition for the existence of componentwise ultimate bounds. This condition is given in terms of the spectral radius of the system matrices which is easy to check and allows us to compute directly both the smallest componentwise ultimate upper bound and the largest componentwise ultimate lower bound. Then, by using the comparison method, we extend the obtained result to a class of nonlinear time-delay systems which has linear positive bounds. Two numerical examples are given to illustrate the effectiveness of the obtained results.